1 Answer
1

Motivic cohomology is an absolute invariant not a geometric one. The projective bundle formula is purely geometric. It reduces the computation of the motivic cohomology of the projective space to that of the base:
$$
H^{p,q}(\mathbb{P}^n_k) = \bigoplus_{i=0}^n H^{p-2i,q-i}(Spec(k))
$$
In general $H^{\bullet,\bullet}(Spec(k))$ is highly non trivial. For example, $H^{1,1}(Spec(k)) = \mathbb{H}^1(Spec(k),\mathbb{G}_m[-1]) = k^\times$ so you will always have $H^{1,1}(\mathbb{P}^n_k) \neq 0$ (except for $k = \mathbb{F}_2$).